scholarly journals A Boundary Integral Equation Related to the Ahlfors Map

2021 ◽  
Vol 17 (2) ◽  
pp. 146-153
Author(s):  
Nur Hazwani Aqilah Abdul Wahid ◽  
Ali Hassan Mohamed Murid ◽  
Mukhiddin I. Muminov
Keyword(s):  
Integral Equation ◽  
Unit Disk ◽  
Boundary Integral Equation ◽  
Mapping Function ◽  
Boundary Integral ◽  
Connected Region ◽  
Multiply Connected ◽  
Ahlfors Map ◽  
Multiply Connected Region

The Ahlfors map is a mapping function that maps a multiply connected region onto a unit disk. This paper presents a new boundary integral equation related to the Ahlfors map of a bounded multiply connected region. The boundary integral equation is constructed from a boundary relationship satisfied by the Ahlfors map of a multiply connected region.

Integral Equation for the Ahlfors Map on Multiply Connected Regions

2015 ◽  
Vol 73 (1) ◽  
Author(s):  
Kashif Nazar ◽  
Ali H. M. Murid ◽  
Ali W. K. Sangawi
Keyword(s):  
Integral Equation ◽  
Boundary Integral Equation ◽  
Boundary Integral ◽  
Connected Region ◽  
Multiply Connected ◽  
Multiple Method ◽  
Boundary Correspondence ◽  
Ahlfors Map ◽  
Multiply Connected Regions ◽  
Multiply Connected Region

This paper presents a new boundary integral equation with the adjoint Neumann kernel associated with  where  is the boundary correspondence function of Ahlfors map of a bounded multiply connected region onto a unit disk. The proposed boundary integral equation is constructed from a boundary relationship satisfied by the Ahlfors map of a multiply connected region. The integral equation is solved numerically for  using combination of Nystrom method, GMRES method, and fast multiple method. From the computed values of    we solve for the boundary correspondence function  which then gives the Ahlfors map. The numerical examples presented here prove the effectiveness of the proposed method.

scholarly journals Analytical Solution for Finding the Second Zero of the Ahlfors Map for an Annulus Region

2019 ◽  
Vol 2019 ◽  
pp. 1-11
Author(s):  
Nur H. A. A. Wahid ◽  
Ali H. M. Murid ◽  
Mukhiddin I. Muminov
Keyword(s):  
Series Representation ◽  
Mapping Function ◽  
Boundary Integral ◽  
Connected Region ◽  
Multiply Connected ◽  
Szegö Kernel ◽  
Szegő Kernel ◽  
Ahlfors Map ◽  
Szego Kernel ◽  
Multiply Connected Region

The Ahlfors map is a conformal mapping function that maps a multiply connected region onto a unit disk. It can be written in terms of the Szegö kernel and the Garabedian kernel. In general, a zero of the Ahlfors map can be freely prescribed in a multiply connected region. The remaining zeros are the zeros of the Szegö kernel. For an annulus region, it is known that the second zero of the Ahlfors map can be computed analytically based on the series representation of the Szegö kernel. This paper presents another analytical method for finding the second zero of the Ahlfors map for an annulus region without using the series approach but using a boundary integral equation and knowledge of intersection points.

scholarly journals Integral and differential equations for conformal mapping of bounded multiply connected regions onto a disk with circular slits

2014 ◽  
Vol 7 (1) ◽  
Author(s):  
Ali H.M. Murid ◽  
Ali W. Kareem Sangawi ◽  
M.M.S. Nasser
Keyword(s):  
Integral Equation ◽  
Conformal Mapping ◽  
Boundary Integral Equation ◽  
Boundary Integral ◽  
Science And Engineering ◽  
Multiply Connected ◽  
Circular Slit ◽  
Multiply Connected Regions ◽  
Multiply Connected Region ◽  
Neumann Kernel

Conformal mapping is a useful tool in science and engineering. On the other hand exact mapping functions are unknown except for some special regions.In this paper we present a new boundary integral equation with classical Neumann kernel associated to f f , where f is a conformal mapping ofbounded multiply connected regions onto a disk with circular slit domain. This boundary integral equation is constructed from a boundary relationshipsatisfied by a function analytic on a multiply connected region. With f f known, one can then treat it as a differential equation for computing f .

scholarly journals Circular Slits Map of Bounded Multiply Connected Regions

2012 ◽  
Vol 2012 ◽  
pp. 1-26 ◽  
Author(s):  
Ali W. K. Sangawi ◽  
Ali H. M. Murid ◽  
M. M. S. Nasser
Keyword(s):  
Integral Equations ◽  
Boundary Integral Equations ◽  
Integral Equation Method ◽  
Boundary Integral ◽  
Connected Region ◽  
Numerical Conformal Mapping ◽  
Multiply Connected ◽  
Circular Slit ◽  
Multiply Connected Regions ◽  
Multiply Connected Region

We present a boundary integral equation method for the numerical conformal mapping of bounded multiply connected region onto a circular slit region. The method is based on some uniquely solvable boundary integral equations with adjoint classical, adjoint generalized, and modified Neumann kernels. These boundary integral equations are constructed from a boundary relationship satisfied by a function analytic on a multiply connected region. Some numerical examples are presented to illustrate the efficiency of the presented method.

scholarly journals Conformal Mapping of Unbounded Multiply Connected Regions onto Canonical Slit Regions

2012 ◽  
Vol 2012 ◽  
pp. 1-29 ◽  
Author(s):  
Arif A. M. Yunus ◽  
Ali H. M. Murid ◽  
Mohamed M. S. Nasser
Keyword(s):  
Integral Equation ◽  
Conformal Mapping ◽  
Integral Equations ◽  
Boundary Integral Equations ◽  
Integral Equation Method ◽  
Boundary Integral ◽  
Linear Boundary ◽  
Multiply Connected ◽  
Multiply Connected Regions ◽  
Multiply Connected Region

We present a boundary integral equation method for conformal mapping of unbounded multiply connected regions onto five types of canonical slit regions. For each canonical region, three linear boundary integral equations are constructed from a boundary relationship satisfied by an analytic function on an unbounded multiply connected region. The integral equations are uniquely solvable. The kernels involved in these integral equations are the modified Neumann kernels and the adjoint generalized Neumann kernels.

scholarly journals A Boundary Integral Equation with the Generalized Neumann Kernel for a Certain Class of Mixed Boundary Value Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Mohamed M. S. Nasser ◽  
Ali H. M. Murid ◽  
Samer A. A. Al-Hatemi
Keyword(s):  
Integral Equation ◽  
Boundary Integral Equation ◽  
Mixed Boundary Condition ◽  
Mixed Boundary ◽  
Boundary Integral ◽  
Numerical Examples ◽  
Multiply Connected ◽  
Generalized Neumann Kernel ◽  
Multiply Connected Regions ◽  
Neumann Kernel

We present a uniquely solvable boundary integral equation with the generalized Neumann kernel for solving two-dimensional Laplace’s equation on multiply connected regions with mixed boundary condition. Two numerical examples are presented to verify the accuracy of the proposed method.

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